The statistical properties of the quasienergy spectrum are used to measure the influence of quantum effects on the quantum kicked rotator which displays chaotic behavior in the classical limit. A transition from orthogonal-ensemble statistics in the semiclassical limit (ℏ→0) to Poisson statistics in the quantum regime is observed. In view of previously obtained results for this system the dependence on the irrationality of ℏ is discussed

Frahm, Holger

Mikeska, H.J.

Institutionelles Repositorium der Leibniz Universität Hannover

VOLUME 60 4 JANUARY 1988 NUMBER 1Quantum Suppression of Irregularity in the Spectral Properties of the Kicked RotatorH. Frahm and H. J. MikeskaInstitut fiir Theoretische Physik, Universitiit Hannover, D 3000 Han-nover I, Federal Republic of Germany(Received 10 August 1987)The statistical properties of the quasienergy spectrum are used to measure the influence of quantumeffects on the quantum kicked rotator which displays chaotic behavior in the classical limit. A transitionfrom orthogonal-ensemble statistics in the semiclassical limit (h 0) to Poisson statistics in the quan-tum regime is observed. In view of previously obtained results for this system the dependence on the ir-rationality of 6 is discussed.PACS numbers: 03.65.—w, 05.45.+bThe statistical properties of the spectra of quantumsystems have been found to be a significant measure forthe degree of integrability of the same system in its clas-sical limit. Using semiclassical arguments, Berry andTabor ' have shown that integrable systems have an un-correlated spectrum showing a Poisson distribution forthe energy separations of adjacent levels. On the otherhand, the theory of random matrices has been appliedsuccessfully to the analysis of the spectral properties ofclassically chaotic systems. In the intermediate re-gime, i.e., for systems that undergo a transition fromnearly integrable to completely chaotic behavior in' theirclassical limit, a transition between these extremal caseshas been found as a function of the classical degree ofstochasticity. ' Thus the statistical properties of thespectrum may be used as a measure for the degree ofregularity in the quantum system.In this Letter we apply this approach to the quantumkicked rotator, introduced by Casati et al. as the basicmodel to study "quantum chaos. " We will present re-sults on how the semiclassical behavior is affected byquantum effects of varying strength, whereas previouswork mostly studied the influence of an increasing degreeof classical stochasticity in the semiclassical limit. Thestatistical properties of the spectrum of the kicked rota-tor have been discussed before by several authors. ' Ac-cording to the work of Feingold and co-workers, thespacing distribution for driven Hamiltonian systems4n+ t =4n + 2ttpn,p„+ t p„+(k/2tt) sing„+ t.(2a)(2b)The kick strength k controls the stochasticity of the stan-dard map. For k 0, the system is nearly integrable;for k = 1 the last Kolmogorov-Arnol'd-Moser torusdisappears and diffusion in the p direction becomes possi-ble.Note that besides the rotational symmetry p' =&+2trand parity conservation (p' = —p, p'= —p), the discretetranslation p'=p+1 does not change the classical dy-namics which defines a (classical) scale of momentutn.This periodicity is a consequence of the special timedependence of the driving field. The correspondingcanonical transformation (p,p) ~ (O,I) which leaves theshould be always Poissonian. We will relate our resultsto this work below.The kicked rotator is a quantized version of the stan-dard map. ' The Hamiltonian of the classical systemreadsp kI0- — cosy'„b(t nT). —2I TBelow we use the dimensionless units, i.e., we assumeI=1, T=2x. The Hamiltonian equations can be in-tegrated over one period of the driving force to give theclassical map (y„denotes the variable y after the nthkick):1987 The American Physical SocietyVOLUME 60, NUMBER 1 PHYSICAL REVIEW LETTERS 4 JANUARY 1988Hamiltonian (1) invariant can be given in terms of agenerating function" which readsF„(y,I,r) =(I+n)(y n—r)+ ,' n —r (3)—(i/h )xp2 i (k/2~A )cosy (4)for the standard map. For any time dependence withbounded spectrum, this translational symmetry in p willnot appear.By application of Floquet's theorem the quantum dy-namics can be described by the discrete time-evolutionoperator~ f 2$U U(2x, 0) =Texp —— diH(r)opwhich can be given explicitly in p representation:U =e ™iJ,(k/2nh ) (5)where p ~ m) = hm ( m) and J„are Bessel functions of thefirst kind. As a result of the discrete set of eigenvalues ofthe operator p, a second momentum scale arises in thequantum system. Depending on whether the classical andthe quantum momentum scales are commensurate ornot, i.e., h [namely the ratio h/(2+I/T) in the originalunits] is rational or irrational, the propagator will have acontinuous (band) or a discrete spectrum. Since wewant to calculate the quasispectrum of U numerically,we will confine ourselves to the case of rational=M/N For .NM even one can reduce U to an NxNmatrix'U (a) —exp imam — i (m——m') —g expW J-iwhere a E [0,2x] is the Bloch number related to thediscrete translational invariance p'=p+M of the quan-tum system. Comparison with the classical system showsthat M is the number of first-order resonances within theperiod of the quantum system. For irrational 6, theclassical periodicity of the system has no quantum ana-log.The statistical properties of the spectrum of a quan-tum system in the semiclassical limit depend on the in-tegrability of the corresponding classical model. A wide-ly used criterion for the characterization of a given spec-trum is the distribution of nearest-neighbor spacings.Berry and Tabor' showed that for a classically integra-ble system this distribution is Poissonian. For classicallychaotic systems it is possible to map the quantum-me-chanical eigenvalue problem to a complicated mechani-cal system whose integrals of motion can be associated—i(m —m') +i cos2' . k 2'+ aN 2@A Nwith the spectral distribution function of the quantumsystem. This procedure gives agreement with certainrandom matrix ensembles: the Gaussian orthogonal en-semble (GOE) for autonomous systems and the circularorthogonal ensemble (COE) for periodically driven sys-tems. For these ensembles the spacing distribution isgiven by the Wigner surmisep(x) = —,' 1rxe (7)to a very good approximation (the spacing x is measuredin units of the mean spacing).For systems which show a transition from regular tochaotic behavior in the classical limit, several interpola-tion formulas for the spacing distribution have been in-troduced (see, e.g., Ref. 5). In this work the one givenby Berry and Robnik' (quoted as BR in the following)will be used:p(x, q) [(1—q) erfc[(Jx/2)qx]+[2q(1 —q)+(n/2)q x]e ' & " }epx+L[n(e) Ae 8] 'de, — —163(L) =—minL A,B "~ (9)where n(e) is the number of states with energy below e.Measuring L in units of the mean level spacing one finds53(L) =L/1 5 for Poisson statistics, (loa)43(L) =1/x lnL —0.007 for OE statistics (L ~ 15).(lob)As before in the case of the level-spacing distribution atransition between the two limiting cases depending onThis gives (7) for q =1 and the Poisson distribution forq=0. Another characteristic statistical quantity is therigidity b,3 which is a measure for correlations in thespectrum:the parameter q can be observed (see, e.g., Ref. 5).To study the level statistics of the kicked-rotator sys-tem, one has to take into account all symmetries of thesystem. A well-defined result can be obtained in the sub-space defined by specific choices of the "good" quantumnumbers parity and Bloch number a only. We analyzedthe spectra of U(a =0) in the even-parity subspace nu-merically for different values of 6 and kick strengthk =10.0. For this value of k the classical phase space iscovered by a single chaotic trajectory to more than 99%.Thus, from semiclassical arguments one expects thequasispectra to obey OE statistics.In Fig. 1 we present the results for the level-spacingdistribution p(x) and the A3 statistics for two differentvalues of h. For I3 = 1/1944, the semiclassical predictionVOLUME 60, NUMBER 1 PHYSICAL REVIEW LETTERS 4 JANUARY 19881.0-1.0 ti0.5—2I40.5—I-3 t-2~og (&j010 20 30 20FIG. 2. BR parameter q from numerical data as a functionof h. The large errors for small values of q are a consequenceof the functional dependence of the BR distribution (8) on q.(a) (b)FIG. 1. Level-spacing distribution and h3 statistics of thecalculated quasienergy spectra of the quantum kicked rotatorfor k -10. (a) h 1/1944 and (b) h 625/1944. Histogramsin the level-spacing diagrams are numerical data, dashed linesare best fits by Eq. (8). Solid lines in the h3 diagrams are Pois-son and OE results (10).of OE-like behavior holds, while for ft =625/1944= 3we find a Poisson distribution of the quasienergies. Onemay use the value of the BR parameter q obtained froma least-squares fit of the numerical results by the BRlevel-spacing distribution function (8) as a measure forthe statistical properties of the quasispectra. Figure 2shows the results of this procedure obtained for q as afunction of tl [We diagonalized U(a =0) in the even-parity subspace for h, =n/1944 which gives 973 levels foreach point. ] In the semiclassical limit ft «1, we find OEproperties of the quasispectrum while for h, -0.1 a tran-sition towards Poissonian distribution is observed. [Di-mensionless values of 5 have a physical meaning in thespecific system for which they have been defined only.To compare our results with different systems one shouldnote that ft ' is the number of (unperturbed) quantumlevels per first-order resonance of the classical system. ]These results are in good agreement with the ones ob-tained earlier for a kicked spin system and the generalobservation that quantum effects tend to suppress theeffects of classical stochasticity. 'It was argued that for irrational h, the statisticalproperties of the spectra should always be Poisson, sincethe eigenvalue problem for U could be mapped on aquasirandom Anderson model' and an exact proof ex-ists' that the energy-level separation in an infinite-dimensional Anderson model obeys Poisson statistics.However, the numerical confirmation of this conjecturesuffers from the fact that there is no possibility to diago-nalize the exact propagation operator in the infinite-dimensional Hilbert space of this system. A truncationof the propagation operator U as given in Eq. (5) is pos-sible by use of the fact that the Bessel functionsJ„(k/2xh) become negligible for n ~ k/xh which givesa band matrix for U. In previous numerical work on thequasispectra of the kicked rotator, this procedure wasused for irrational ft = 1 where Poissonian distributionsof the eigenvalues have been found. This result was in-terpreted as a confirmation of the equivalence of drivenHamiltonian quantum systems with the Anderson model.From our results it is clear that this has to be expectedfor values of the quantum parameter lt = 1 independentof the classical properties of the system. The crucialquestion is whether this conclusion can also be drawn ifone approaches the semiclassical limit. Our results indi-cate that a transition to Wigner statistics occurs atsufficiently small values of h.To investigate this question further, we have calculat-ed the quasispectrum for h«1 (where the truncationprocedure is not applicable) by using a rational M/N=6 with M, N being relatively prime. For large N thewidth of the quasienergy bands decreases exponentiallyand the dependence on the Bloch number a may beneglected. 'To prove that for l't 0 the spectral properties againchange to OE-like behavior, we investigated the spectraof U(a) for rational approximations to the most irration-al value of l't near ho~ 1/100 which may be obtained byadding 1 to the continued-fraction (CF) expansion" ofhp =1/N:—= [N, 1, 1, . . .].1+ I+. . .(The dependence of the time evolution of the quantumkicked rotator on the irrationality of 6 as measured byan increasing length of the CF was investigated recentlyby Casati et al. ' ) Table I shows the dependence of theBR parameter q on the length of this expansion wherefor technical reasons different N in the range of 103 to106 were chosen. A dependence on the "irrationality" ofVOLUME 60, NUMBER 1 PHYSICAL REVIEW LETTERS 4 JANUARY 1988TABLE I. Dependence of the BR parameter q on the irra-tionality of h as measured by the length of the CF expansionof h.3/3205/52813/136021/2176CF (h)[106,1,1,1][105,1,1,1,1l[104,1,1,1,1,1,1l[103,1,1,1,1,1,1,1]1.0000.9890.9951.000+ 0.204~ 0.133+ 0.059~ 0.085its has not been found in the range of parameters we wereable to treat numerically (up to l1 21/2176 which re-quires the diagonalization of a unitary 1089& 1089 ma-trix). However, any rational value M/N of h corre-sponds to a pseudorandom lattice with N sites instead ofan infinite system and we are not able to show thatWigner statistics persists in the limit N ~ withM/N fixed. Therefore the results of the present pa-per are consistent with those of Ref. 9 if a transition toPoisson statistics appears in this limit.From the results presented above we conclude thatquantum effects can drastically change the spectral prop-erties of systems which are classically chaotic. Theinfluence of the irregular character of the correspondingclassical system which is well established in the semiclas-sical regime disappears. The observed transition of thespectral characteristics as a function of l't is analogous tothe one found in semiclassical systems (ft 0) when theparameter that controls the classical stochasticity —k inthe standard map —is varied. Both effects can be seen asdifferent aspects of the same phenomenon, namely, thedegree of irregularity in a quantum system. It is propor-tional to the classical one in the semiclassical regimewhile it vanishes for l't 1, i.e., classical and quantumirregularity become independent. If measured by the BRparameter q it should be possible to describe this quan-tum irregularity by a universal scaling function q(k, h).However, a theory which describes this scaling has to gobeyond semiclassical arguments and is not available sofar.In the investigation of the spectral characteristics intheir dependence on the number-theoretical properties ofl't we have not been able to verify the conjecture thatthere should be a transition towards Poisson statisticswhen one approaches irrational values of 6 (Table I). Itcan be argued that this transition occurs for continued-fraction expansion of h even longer than the ones con-sidered in this work. However, one should rememberthat the difference in the quantum dynamics between ra-tional and irrational values of A, is a consequence of theclassical periodicity of the standard map in p which wasgenerated by choice of a driving force with unboundedspectrum. Thus, a more realistic system will not showthis phenomenon.One of us the authors (H.F.) wishes to thank Dr. B.Eckhardt, Dr. R. Schraf, and Dr. T. Dittrich for interest-ing discussions and useful comments. We also want tothank Professor R. E. Prange and Professor F. Haake forcorrespondence on an earlier version of the manuscript.We gratefully acknowledge the cooperation of the Re-gionales Rechenzentrum fiir Niedersachsen, Hannover,where the numerical calculations were performed.'M. V. Berry and M. Tabor, Proc. Roy. Soc. London, Ser. A356, 375 (1977).2P. Pechukas, Phys. Rev. Lett. 51, 943 (1983); T. Yukawa,Phys. Rev. Lett. 54, 1883 (1985), and Phys. Lett. A 116, 2271986).F. Haake, M. Kus, and R. Scharf, Z. Phys. B 65, 381(1987); K. Nakamura and H. J. Mikeska, Phys. Rev. A 35,5294 (1987).4E. Hailer, H. Koppel, and L. S. Cederbaum, Phys. Rev.Lett. 52, 1665 (1984); T. H. Seligman, J. J. M. Verbaatschot,and M. R. Zirnbauer, Phys. Rev. Lett. 53, 215 (1984), and J.Phys. A 18, 2751 (1985).sH. Frahm and H. J. Mikeska, Z. Phys. B 65, 249 (1986).G. Casati, B. V. Chirikov, F. M. Izrailev, and J. Ford, inStochastic Behauior in Classical and Quantum HamiltonianSystems, edited by G. Casati and J. Ford, Lecture Notes inPhysics Vol 93 (Springer-Verlag, New York, 1979).7The quantum kicked rotator for varying values of h hasbeen discussed by T. Geisel, G. Radons, and J. Rubner, Phys.Rev. Lett. 57, 2883 (1986); the dependence of the level statis-tics on h has been discussed for different models in Ref. 5 andby J. V. Jose and R. Cordery, Phys. Rev. Lett. 56, 290 (1986).The spectrum of the kicked rotator for fixed rational 6 butvery large k was investigated by F. M. Izraelev, Phys. Rev.Lett. 56, 541 (1986).M. Feingold, S. Fishman, D. R. Grempel, and R. E. Prange,Phys. Rev. B 31, 6852 (1985); M. Feingold and S. Fishman,Physica (Amsterdam) 25D, 181 (1987).'OB. V. Chirikov, Phys. Rep. 52, 263 (1979).A. J. Lichtenberg and M. A. Lieberman, Regular and Sto-chastic Motion (Springer-Verlag, New York, 1983).12S. J. Chang and K. J. Chi, Phys. Rev. A 34, 7 (1986).'sM. V. Berry and M. Robnik, J. Phys. A 17, 2413 (1984).t4H. Frahm and H. J. Mikeska, Z. Phys. B 60, 117 (1985).S. Fishman, D. R. Grempel, and R. E. Prange, Phys. Rev.Lett. 49, 509 (1982), and Phys. Rcv. A 29, 1639 (1984).'sS. A. Molcanov, Commun. Math. Phys. 7$, 429 1981).G. Casati, J. Ford, I. Guarneri, and F. Vivaldi, Phys. Rev.A 34, 1413 (1986).

Quantum suppression of irregularity in the spectral properties of the kicked rotator

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Quantum suppression of irregularity in the spectral properties of the kicked rotator

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